
Post by Elizabeth on Aug 8, 2020 5:57:23 GMT
Heard this recently from someone who likes philosophy and loved the quote. What are your thoughts of it?
To search for God with logical proof .. is like searching for the Sun with a Lamp. Sufi poverb
I think it makes all the sense in the world. Good job Sufi...whoever/whatever you are. I say this because people are searching for it the wrong way. Turn the lamp off and find the light of the sun. Turn off the way you think your logic works and find how things truly work.



Post by Eugene 2.0 on Aug 19, 2020 16:55:33 GMT
karlEnumerating means algorithmization? Isn't the algorithm set of instructions, like a) go right b) pick up keys c) uf there's no keys, go left... I thought that formalization of the algorithm makes us impossible to prove the incompleteness. And, the set of all numbers minus the complementary set equals an enumerated set?



Post by karl on Aug 19, 2020 17:08:33 GMT
karl Enumerating means algorithmization? Isn't the algorithm set of instructions, like a) go right b) pick up keys c) uf there's no keys, go left... I thought that formalization of the algorithm makes us impossible to prove the incompleteness. And, the set of all numbers minus the complementary set equals an enumerated set?
Imagine you have a set A which is a subset of natural numbers. If it's possible to write an algorithm to produce that set, then that means the set is enumerable.
If you mean the complementary set I elaborated on in the post, yes.
Set 1: An undecidable, enumerable set Set 2: The complementary of Set 1.
Set of all natural numbers minus set 2=set 1, which is an enumerable set.



Post by Eugene 2.0 on Aug 19, 2020 17:19:40 GMT
karl Enumerating means algorithmization? Isn't the algorithm set of instructions, like a) go right b) pick up keys c) uf there's no keys, go left... I thought that formalization of the algorithm makes us impossible to prove the incompleteness. And, the set of all numbers minus the complementary set equals an enumerated set?
Imagine you have a set A which is a subset of natural numbers. If it's possible to write an algorithm to produce that set, then that means the set is enumerable.
If you mean the complementary set I elaborated on in the post, yes.
Set 1: An undecidable, enumerable set Set 2: The complementary of Set 1.
Set of all natural numbers minus set 2=set 1, which is an enumerable set.
You're explaining awesomely. I hope logic of minusing sets I got, but I still don't understand why the natural numbers = algorithm? A man is walking and sees a piece of paper, then he grabs it and reads. It says: do blahblahblah, then do blahblahblah, and then do blahblahblah. There's no any natural numbers, and nevertheless paper has instructions, i.e. the algorithm. Also, if a set S is complementary to a certain set of all natural numbers, then why this S is undecidable? The natural numers → decidable?



Post by karl on Aug 19, 2020 17:35:33 GMT
Imagine you have a set A which is a subset of natural numbers. If it's possible to write an algorithm to produce that set, then that means the set is enumerable.
If you mean the complementary set I elaborated on in the post, yes.
Set 1: An undecidable, enumerable set Set 2: The complementary of Set 1.
Set of all natural numbers minus set 2=set 1, which is an enumerable set.
You're explaining awesomely. I hope logic of minusing sets I got, but I still don't understand why the natural numbers = algorithm? A man is walking and sees a piece of paper, then he grabs it and reads. It says: do blahblahblah, then do blahblahblah, and then do blahblahblah. There's no any natural numbers, and nevertheless paper has instructions, i.e. the algorithm. Also, if a set S is complementary to a certain set of all natural numbers, then why this S is undecidable? The natural numers → decidable?
I'm referring to a computer algorithm that produces a set of natural numbers. Just imagine that you start a software on your computer, it flips open a window and starts churning out a bunch of numbers. That's an example of an algorithm that produces a set of numbers.
The proof for the incompleteness theorem shows that there exists an enumerable but undecidable subset of natural numbers. From that we can deduce that the complementary of that set is unenumerable.
Why that particular set is undecidable is something I can explain, but it would have to be a very long explanation. The proof I read leading to it was outlined over 20 pages. You can find it here:
You wrote: "a set S is complementary to a certain set of all natural numbers"
I presume you meant subset. The complementary to a set of all natural numbers is an empty set. If you have a subset of natural numbers, then it is NOT given that its complementary S is undecidable. It may or it may not be. But IF S is undecidable, then its complementary is unenumerable.



Post by Eugene 2.0 on Aug 19, 2020 19:11:33 GMT
You're explaining awesomely. I hope logic of minusing sets I got, but I still don't understand why the natural numbers = algorithm? A man is walking and sees a piece of paper, then he grabs it and reads. It says: do blahblahblah, then do blahblahblah, and then do blahblahblah. There's no any natural numbers, and nevertheless paper has instructions, i.e. the algorithm. Also, if a set S is complementary to a certain set of all natural numbers, then why this S is undecidable? The natural numers → decidable?
I'm referring to a computer algorithm that produces a set of natural numbers. Just imagine that you start a software on your computer, it flips open a window and starts churning out a bunch of numbers. That's an example of an algorithm that produces a set of numbers.
The proof for the incompleteness theorem shows that there exists an enumerable but undecidable subset of natural numbers. From that we can deduce that the complementary of that set is unenumerable.
Why that particular set is undecidable is something I can explain, but it would have to be a very long explanation. The proof I read leading to it was outlined over 20 pages. You can find it here:
You wrote: "a set S is complementary to a certain set of all natural numbers"
I presume you meant subset. The complementary to a set of all natural numbers is an empty set. If you have a subset of natural numbers, then it is NOT given that its complementary S is undecidable. It may or it may not be. But IF S is undecidable, then its complementary is unenumerable.
Thanks for the link. I thought Uspensky's explanation was difficult enough to get it for not a mathematician. Oh, I get what is "complementary". If A is a set, than A`, or ~A, or A ^{c} is complementary. Isn't it the same is an addition to a set. Anyway, so to solve any formal theorem it should be written numerically, being coded into Godel's number system, and only then we should look up for an algorithm, right? Quite strangely, isn't it? It's like to rewrite some phrases into another language, and them check 'em to correspondence to syntax, grammar, style or smth like that. For example, coding violas into drums. It's possible, but without a doubt in such a success it would have many lacks (it would be sounds too poorly to convey a sound or, in general, the sense. Discreetness can't convey an actual sound of smooth, linear, continuality. And also, why a subset of all natural numbers should be without an algorithm? There are so plenty prohibitions: not to take algorithms without its numerical condition; write algorithms just for infinite numbers... Anyway, the theorem is interesting and, without any doubts, powerful. If you said that our souls ('s imagination) were wider than any other landscapes (brain landscapes), then the Godel's proof was beyond all it. It proved a part, not the soul's field. And it proved not even mind's territory. Because of narrowing we didn't get too much about this. I guess.



Post by karl on Aug 19, 2020 20:18:07 GMT
I'm referring to a computer algorithm that produces a set of natural numbers. Just imagine that you start a software on your computer, it flips open a window and starts churning out a bunch of numbers. That's an example of an algorithm that produces a set of numbers.
The proof for the incompleteness theorem shows that there exists an enumerable but undecidable subset of natural numbers. From that we can deduce that the complementary of that set is unenumerable.
Why that particular set is undecidable is something I can explain, but it would have to be a very long explanation. The proof I read leading to it was outlined over 20 pages. You can find it here:
You wrote: "a set S is complementary to a certain set of all natural numbers"
I presume you meant subset. The complementary to a set of all natural numbers is an empty set. If you have a subset of natural numbers, then it is NOT given that its complementary S is undecidable. It may or it may not be. But IF S is undecidable, then its complementary is unenumerable.
Thanks for the link. I thought Uspensky's explanation was difficult enough to get it for not a mathematician. Oh, I get what is "complementary". If A is a set, than A`, or ~A, or A ^{c} is complementary. Isn't it the same is an addition to a set. Anyway, so to solve any formal theorem it should be written numerically, being coded into Godel's number system, and only then we should look up for an algorithm, right? Quite strangely, isn't it? It's like to rewrite some phrases into another language, and them check 'em to correspondence to syntax, grammar, style or smth like that. For example, coding violas into drums. It's possible, but without a doubt in such a success it would have many lacks (it would be sounds too poorly to convey a sound or, in general, the sense. Discreetness can't convey an actual sound of smooth, linear, continuality. And also, why a subset of all natural numbers should be without an algorithm? There are so plenty prohibitions: not to take algorithms without its numerical condition; write algorithms just for infinite numbers... Anyway, the theorem is interesting and, without any doubts, powerful. If you said that our souls ('s imagination) were wider than any other landscapes (brain landscapes), then the Godel's proof was beyond all it. It proved a part, not the soul's field. And it proved not even mind's territory. Because of narrowing we didn't get too much about this. I guess.
Just to clarify what a complementary set is:
Let's say set A is the set of all even numbers: 0, 2, 4, 6, 8.... Then set A is a subset of natural numbers.
(All natural numbers = 0, 1, 2, 4, 5, 6, 7, 8, 9...)
The let's say set B is the set of all odd numbers: 1, 3, 5, 7, 9.... Then set B is the complementary set of set A.
All natural numbers minus the set A = set B All natural numbers minus the set B = set A
As for the incompleteness theorem. Yes, if a theorem is deducible from given axioms in an arithmetic system, then it's possible to write it as a number sequence. If a theorem is correct within that system, but not deducible, then an axiom has to be added to make it deducible. For example, the theorem itself may be added as an axiom. But no matter how many axioms one adds, there will always be theorems that are true but unprovable within that system.
You wrote: "why a subset of all natural numbers should be without an algorithm?"
If you mean to ask: "Why does there exist a subset of natural numbers that can't be produced by an algorithm?"
Then that would require too much text to explain accurately. (Uspensky explained it over 26 pages.) But I can offer my own way of looking at it.
Imagine that there exists some kind of mysterious device that is capable of producing an unenumerable set. As you read the numbers it produces, you'll notice that no matter how hard you try, you can't find any pattern in them. And if you do think you've found one, the pattern will sooner or later break, as more numbers are added. This will make you think that the sequence is random. But it's not random. It's clearly defined. It's just that what defines it makes it infinitely complex, which is why no pattern can be recognised. If there was a pattern, its complexity would hence be limited. Why does there exist subsets that contain an infinite amount of information? It's due to the complexity of the system itself. In arithmetic that only allows for adding and subtracting, Gödel's theorem doesn't hold. The reason for this is that the system is so simple, it doesn't allow for anything infinitely complex. But the arithmetic system, which includes multiplication and addition, adds complexity, and this complexity of the system allows for infinitely complex mathematical entities to exist within the system. In a sense, the system becomes so complex it can't handle its own complexity.



Post by Eugene 2.0 on Aug 19, 2020 20:33:40 GMT
Thanks for the link. I thought Uspensky's explanation was difficult enough to get it for not a mathematician. Oh, I get what is "complementary". If A is a set, than A`, or ~A, or A ^{c} is complementary. Isn't it the same is an addition to a set. Anyway, so to solve any formal theorem it should be written numerically, being coded into Godel's number system, and only then we should look up for an algorithm, right? Quite strangely, isn't it? It's like to rewrite some phrases into another language, and them check 'em to correspondence to syntax, grammar, style or smth like that. For example, coding violas into drums. It's possible, but without a doubt in such a success it would have many lacks (it would be sounds too poorly to convey a sound or, in general, the sense. Discreetness can't convey an actual sound of smooth, linear, continuality. And also, why a subset of all natural numbers should be without an algorithm? There are so plenty prohibitions: not to take algorithms without its numerical condition; write algorithms just for infinite numbers... Anyway, the theorem is interesting and, without any doubts, powerful. If you said that our souls ('s imagination) were wider than any other landscapes (brain landscapes), then the Godel's proof was beyond all it. It proved a part, not the soul's field. And it proved not even mind's territory. Because of narrowing we didn't get too much about this. I guess.
Just to clarify what a complementary set is:
Let's say set A is the set of all even numbers: 0, 2, 4, 6, 8.... Then set A is a subset of natural numbers.
(All natural numbers = 0, 1, 2, 4, 5, 6, 7, 8, 9...)
The let's say set B is the set of all odd numbers: 1, 3, 5, 7, 9.... Then set B is the complementary set of set A.
All natural numbers minus the set A = set B All natural numbers minus the set B = set A
As for the incompleteness theorem. Yes, if a theorem is deducible from given axioms in an arithmetic system, then it's possible to write it as a number sequence. If a theorem is correct within that system, but not deducible, then an axiom has to be added to make it deducible. For example, the theorem itself may be added as an axiom. But no matter how many axioms one adds, there will always be theorems that are true but unprovable within that system.
You wrote: "why a subset of all natural numbers should be without an algorithm?"
If you mean to ask: "Why does there exist a subset of natural numbers that can't be produced by an algorithm?"
Then that would require too much text to explain accurately. (Uspensky explained it over 26 pages.) But I can offer my own way of looking at it.
Imagine that there exists some kind of mysterious device that is capable of producing an unenumerable set. As you read the numbers it produces, you'll notice that no matter how hard you try, you can't find any pattern in them. And if you do think you've found one, the pattern will sooner or later break, as more numbers are added. This will make you think that the sequence is random. But it's not random. It's clearly defined. It's just that what defines it makes it infinitely complex, which is why no pattern can be recognised. If there was a pattern, its complexity would hence be limited. Why does there exist subsets that contain an infinite amount of information? It's due to the complexity of the system itself. In arithmetic that only allows for adding and subtracting, Gödel's theorem doesn't hold. The reason for this is that the system is so simple, it doesn't allow for anything infinitely complex. But the arithmetic system, which includes multiplication and addition, adds complexity, and this complexity of the system allows for infinitely complex mathematical entities to exist within the system. In a sense, the system becomes so complex it can't handle its own complexity.
Karl, I confirm it again  how brilliant your explanations are! Now I see it. No, I didn't mean that  about a subset, it's ok. I meant why what is not numerical (i.e. doesn't belong to the natural numbers  and you said that the natural numbers and algorithms are together, because we can watch it as on a computer's monitor) can't be algorithmized? And now you said that what is derived from axioms and somehow relates (connected?) to arithmetics (summing, adding, multiplying...?), then it can be coded. The mechanism of coding I'll try to get by myself; I just want to understand ehy it's not possible to algorithmize something that doesn't have any numbers.



Post by karl on Aug 19, 2020 21:45:08 GMT
Just to clarify what a complementary set is:
Let's say set A is the set of all even numbers: 0, 2, 4, 6, 8.... Then set A is a subset of natural numbers.
(All natural numbers = 0, 1, 2, 4, 5, 6, 7, 8, 9...)
The let's say set B is the set of all odd numbers: 1, 3, 5, 7, 9.... Then set B is the complementary set of set A.
All natural numbers minus the set A = set B All natural numbers minus the set B = set A
As for the incompleteness theorem. Yes, if a theorem is deducible from given axioms in an arithmetic system, then it's possible to write it as a number sequence. If a theorem is correct within that system, but not deducible, then an axiom has to be added to make it deducible. For example, the theorem itself may be added as an axiom. But no matter how many axioms one adds, there will always be theorems that are true but unprovable within that system.
You wrote: "why a subset of all natural numbers should be without an algorithm?"
If you mean to ask: "Why does there exist a subset of natural numbers that can't be produced by an algorithm?"
Then that would require too much text to explain accurately. (Uspensky explained it over 26 pages.) But I can offer my own way of looking at it.
Imagine that there exists some kind of mysterious device that is capable of producing an unenumerable set. As you read the numbers it produces, you'll notice that no matter how hard you try, you can't find any pattern in them. And if you do think you've found one, the pattern will sooner or later break, as more numbers are added. This will make you think that the sequence is random. But it's not random. It's clearly defined. It's just that what defines it makes it infinitely complex, which is why no pattern can be recognised. If there was a pattern, its complexity would hence be limited. Why does there exist subsets that contain an infinite amount of information? It's due to the complexity of the system itself. In arithmetic that only allows for adding and subtracting, Gödel's theorem doesn't hold. The reason for this is that the system is so simple, it doesn't allow for anything infinitely complex. But the arithmetic system, which includes multiplication and addition, adds complexity, and this complexity of the system allows for infinitely complex mathematical entities to exist within the system. In a sense, the system becomes so complex it can't handle its own complexity.
Karl, I confirm it again  how brilliant your explanations are! Now I see it. No, I didn't mean that  about a subset, it's ok. I meant why what is not numerical (i.e. doesn't belong to the natural numbers  and you said that the natural numbers and algorithms are together, because we can watch it as on a computer's monitor) can't be algorithmized? And now you said that what is derived from axioms and somehow relates (connected?) to arithmetics (summing, adding, multiplying...?), then it can be coded. The mechanism of coding I'll try to get by myself; I just want to understand ehy it's not possible to algorithmize something that doesn't have any numbers.
Thank you.
This discussion about algorithms started by me writing that if a set is enumerable, it can be expressed by an algorithm. And by algorithm I was referring to the type of algorithms that one finds in computer languages.
However, if one want, one may give the term "algorithm" a wider definition, and by wikipedia it's "a process or set of rules to be followed in calculations or other problemsolving operations, especially by a computer." By that definition, your earlier example could be included:
a) go right b) pick up keys c) uf there's no keys, go left..." And if that algorithm was to be executed by a robot, it would have to be written as a computer algorithm. But that would not be necessary if these are just instructions given to a human.



Post by Eugene 2.0 on Aug 19, 2020 23:10:45 GMT
Karl, I confirm it again  how brilliant your explanations are! Now I see it. No, I didn't mean that  about a subset, it's ok. I meant why what is not numerical (i.e. doesn't belong to the natural numbers  and you said that the natural numbers and algorithms are together, because we can watch it as on a computer's monitor) can't be algorithmized? And now you said that what is derived from axioms and somehow relates (connected?) to arithmetics (summing, adding, multiplying...?), then it can be coded. The mechanism of coding I'll try to get by myself; I just want to understand ehy it's not possible to algorithmize something that doesn't have any numbers.
Thank you.
This discussion about algorithms started by me writing that if a set is enumerable, it can be expressed by an algorithm. And by algorithm I was referring to the type of algorithms that one finds in computer languages.
However, if one want, one may give the term "algorithm" a wider definition, and by wikipedia it's "a process or set of rules to be followed in calculations or other problemsolving operations, especially by a computer." By that definition, your earlier example could be included:
a) go right b) pick up keys c) uf there's no keys, go left..." And if that algorithm was to be executed by a robot, it would have to be written as a computer algorithm. But that would not be necessary if these are just instructions given to a human. Aha, I got it. Yes, I was quite surprised when I read that the algorithm was what was tied up with natural numbers. I thought Uspensky's short book contains it  the answer to this question. Now I see. And this time I am impressed even more – if the number coding uses to translate it (some texts, instructions) to the neutral language (computer ones are supposed to be these), then incompleteness may show inabilities of AI to understand something. Do you think the same? The free will then is human only (i.e. and such creatures that alike us) ability? I'd like to ask you another question if you don't mind. I wanted to buy Uspensky book (about the incompleteness) years ago, and it seemed to me too tough. Now I see it as difficult too. However, I wanted to try it. Many similar books, like Kleene's math logic (1970), Tarski's exercisebook (1948), and Slupecki, and Borkoeski Logic and Sets (1950's?) are much merciful to their readers, because they step by step explain the rules and etc. Recently I discovered Landau's "Basics of Analysis" that was the same  completely explicit and detailed. I noticed that for me is really important that stepbystep explanation that I can repeat by myself. So, my question is: how did you read Uspensky's article? And did you read the Gödel's proof from there or from the other sources? (If there're reasons why you don't want to answer, it's ok. I don't insist on it, I'm really interested in.) What do you think about the algorithm of finding God? Could such an algorithm be found? There was a book... uhh, I think it's A. Asimov's one where at the Eastern monastery some monks searched for Gods names, and they had found it occasionally. Have you heard about this story?



Post by karl on Aug 20, 2020 10:49:27 GMT
Thank you.
This discussion about algorithms started by me writing that if a set is enumerable, it can be expressed by an algorithm. And by algorithm I was referring to the type of algorithms that one finds in computer languages.
However, if one want, one may give the term "algorithm" a wider definition, and by wikipedia it's "a process or set of rules to be followed in calculations or other problemsolving operations, especially by a computer." By that definition, your earlier example could be included:
a) go right b) pick up keys c) uf there's no keys, go left..." And if that algorithm was to be executed by a robot, it would have to be written as a computer algorithm. But that would not be necessary if these are just instructions given to a human. Aha, I got it. Yes, I was quite surprised when I read that the algorithm was what was tied up with natural numbers. I thought Uspensky's short book contains it  the answer to this question. Now I see. And this time I am impressed even more – if the number coding uses to translate it (some texts, instructions) to the neutral language (computer ones are supposed to be these), then incompleteness may show inabilities of AI to understand something. Do you think the same? The free will then is human only (i.e. and such creatures that alike us) ability? I'd like to ask you another question if you don't mind. I wanted to buy Uspensky book (about the incompleteness) years ago, and it seemed to me too tough. Now I see it as difficult too. However, I wanted to try it. Many similar books, like Kleene's math logic (1970), Tarski's exercisebook (1948), and Slupecki, and Borkoeski Logic and Sets (1950's?) are much merciful to their readers, because they step by step explain the rules and etc. Recently I discovered Landau's "Basics of Analysis" that was the same  completely explicit and detailed. I noticed that for me is really important that stepbystep explanation that I can repeat by myself. So, my question is: how did you read Uspensky's article? And did you read the Gödel's proof from there or from the other sources? (If there're reasons why you don't want to answer, it's ok. I don't insist on it, I'm really interested in.) What do you think about the algorithm of finding God? Could such an algorithm be found? There was a book... uhh, I think it's A. Asimov's one where at the Eastern monastery some monks searched for Gods names, and they had found it occasionally. Have you heard about this story?
It was from Uspensky's book I first learned the proof for the incompleteness theorem. And yes, it's quite difficult reading. It's immensely abstract.
Yes, I think free will depends on conscious intelligence. And I do believe human intelligence is something substantially beyond AI.
Imagine that you invent a computer language. Then you write an algorithm for enumerating every possible combinations of symbols within that language. So each combination will be given an index number, starting from "1". Most of those combinations will be complete rubbish, but a tiny fraction of them will constitute a meaningful algorithm. And since each algorithm has an index number, that set of index numbers, which we shall call set A, is a subset of natural numbers. According to one axiom for Gödel's incompleteness theorem, it's possible to write an algorithm to identify all meaningful algorithms within that computer language. This means that set A is enumerable. But let's look for a subgroup of set A, which we shall call set B. Set B has a particular property which we shall call Z. Z is the property of that if you run the algorithm, it will produce a never ending sequence of digits. It doesn't specify which digits. It could be just an endless series of 1's. Or it could be the decimals of PI.
Now the question is: Does there exist an algorithm to identify all members of set B? No, it's possible to prove that there does not. This also means that there exists no algorithm to produce the set of index numbers for each member of set B. Or, in other words, set B is an unenumerable set.
But what if you, the creator of the computer language, decided to just go through the algorithms of set A, one by one, to identify which of them had property Z. And when you found one, you'd write down its index number. If you were able to do that, you'd be, in effect, producing an unenumerable set. Something no computer can do.
I don't believe in an algorithm to find God. I think God is found intuitively.



Post by Eugene 2.0 on Aug 20, 2020 13:24:12 GMT
Aha, I got it. Yes, I was quite surprised when I read that the algorithm was what was tied up with natural numbers. I thought Uspensky's short book contains it  the answer to this question. Now I see. And this time I am impressed even more – if the number coding uses to translate it (some texts, instructions) to the neutral language (computer ones are supposed to be these), then incompleteness may show inabilities of AI to understand something. Do you think the same? The free will then is human only (i.e. and such creatures that alike us) ability? I'd like to ask you another question if you don't mind. I wanted to buy Uspensky book (about the incompleteness) years ago, and it seemed to me too tough. Now I see it as difficult too. However, I wanted to try it. Many similar books, like Kleene's math logic (1970), Tarski's exercisebook (1948), and Slupecki, and Borkoeski Logic and Sets (1950's?) are much merciful to their readers, because they step by step explain the rules and etc. Recently I discovered Landau's "Basics of Analysis" that was the same  completely explicit and detailed. I noticed that for me is really important that stepbystep explanation that I can repeat by myself. So, my question is: how did you read Uspensky's article? And did you read the Gödel's proof from there or from the other sources? (If there're reasons why you don't want to answer, it's ok. I don't insist on it, I'm really interested in.) What do you think about the algorithm of finding God? Could such an algorithm be found? There was a book... uhh, I think it's A. Asimov's one where at the Eastern monastery some monks searched for Gods names, and they had found it occasionally. Have you heard about this story?
It was from Uspensky's book I first learned the proof for the incompleteness theorem. And yes, it's quite difficult reading. It's immensely abstract.
Yes, I think free will depends on conscious intelligence. And I do believe human intelligence is something substantially beyond AI.
Imagine that you invent a computer language. Then you write an algorithm for enumerating every possible combinations of symbols within that language. So each combination will be given an index number, starting from "1". Most of those combinations will be complete rubbish, but a tiny fraction of them will constitute a meaningful algorithm. And since each algorithm has an index number, that set of index numbers, which we shall call set A, is a subset of natural numbers. According to one axiom for Gödel's incompleteness theorem, it's possible to write an algorithm to identify all meaningful algorithms within that computer language. This means that set A is enumerable. But let's look for a subgroup of set A, which we shall call set B. Set B has a particular property which we shall call Z. Z is the property of that if you run the algorithm, it will produce a never ending sequence of digits. It doesn't specify which digits. It could be just an endless series of 1's. Or it could be the decimals of PI.
Now the question is: Does there exist an algorithm to identify all members of set B? No, it's possible to prove that there does not. This also means that there exists no algorithm to produce the set of index numbers for each member of set B. Or, in other words, set B is an unenumerable set.
But what if you, the creator of the computer language, decided to just go through the algorithms of set A, one by one, to identify which of them had property Z. And when you found one, you'd write down its index number. If you were able to do that, you'd be, in effect, producing an unenumerable set. Something no computer can do.
I don't believe in an algorithm to find God. I think God is found intuitively.
Hmm.. Even if a computer would be able to draw anything that would not be digits (some artistic computers), and even if they would get an ability to deciphe one's drawings, it would not allow them complete their "thinking". First of all, why do we need thinking that to what the invented computer algorithmizing some sets of symbols (let it be Egyptian mystical messages) has index numbers? Doesn't it depend on the computer? We consider a computer to be a logical, the sequential type system, but what makes us stop taking the ones as based on mechanical principles? Surely, I can't say I know what kind of technology it must be. If the computer takes the Egyptian scriptures and converting it into language L in which each string has a sign (the mark) that this string is converted by a computer (as for me, it's the same as to say that a computer can recognise the string as a string). Then it codes it into another language that can be understood by a man or, at least that a man can hope to understand. I know that this fat speculation leads to the question about languages and how translate one into another, but don't usually say it about our languages since there's always something that help us to do it... I guess it might be a soul or something beyond our presented "human shape". As you said  the intuition. And also, what makes B to have such Z that gives us infinite lines? Isn't it the complementary set (to natural numbers) you've been mentioning? Intuition must be a positive property. Because it allows to find God not studying hard for mastering difficult logical structures and forms. Since that God doesn't require any complex studies to be able to comprehended or heard Him, and, in turn, no need to try to prove His existence.



Post by karl on Aug 20, 2020 14:16:59 GMT
You wrote: "what makes B to have such Z that gives us infinite lines?"
Eh... I don't understand the meaning of that question. I simply defined set B as having property Z. And the property Z is to produce an infinite sequence of digits. I didn't mention geometric lines.
If you run a software with property Z on your computer, it would start churning out digits. For example, looking something like this:
"0002340356705020520304570676574506304502350203503525235230457067805065407405634062052034503250324523..."
And go on forever. The question is: If you present an algorithm to an AI and ask: "Does this algorithm have property Z?", then it cannot be trusted to get the answer right for every algorithm it's presented with.



Post by Eugene 2.0 on Aug 20, 2020 14:35:27 GMT
You wrote: "what makes B to have such Z that gives us infinite lines?" Eh... I don't understand the meaning of that question. I simply defined set B as having property Z. And the property Z is to produce an infinite sequence of digits. I didn't mention geometric lines. If you run a software with property Z on your computer, it would start churning out digits. For example, looking something like this: "0002340356705020520304570676574506304502350203503525235230457067805065407405634062052034503250324523..." And go on forever. The question is: If you present an algorithm to an AI and ask: "Does this algorithm have property Z?", then it cannot be trusted to get the answer right for every algorithm it's presented with. It exactly what I asked. Ok. Also, I'd like to correct my previous comment I said incorrectly. I asked whether natural numbers had contemplate set, however I didn't ask about of even or odd, but about the other ones (are there any?). That B can have different properties, right? Is it necessary for A to have B – the subset that contains Z? – What is my interest in here? What about such a group of symbols of a certain language (e.g. some mystery Egyptian messages) that can be perfectly translated without that Z. Is it possible? Do all the groups of objects that are being coded (by a computer) have that set Z? And "Z" I meant here as the set that produces such enumerated combinations of symbols.



Post by Eugene 2.0 on Aug 20, 2020 14:42:02 GMT
karlAlso, "Z" cannot be solved by a computer (it's incomputable), because it cannot imply that Z is presented there (right?). I mean in one of previous comments you said that a person, a human being, is able to try to put such enumerated algorithm, but then that algorithm wouldn't be numerated, i.e. computable. And that is why I started to ask myself if a person was the one who was able to find that Z? Defining that "there is such Z..." we already try to get the sense of it  from that formula. And we can name such a set is enumerated, but a computer doesn't (?).



Post by karl on Aug 20, 2020 16:08:09 GMT
You wrote: "what makes B to have such Z that gives us infinite lines?" Eh... I don't understand the meaning of that question. I simply defined set B as having property Z. And the property Z is to produce an infinite sequence of digits. I didn't mention geometric lines. If you run a software with property Z on your computer, it would start churning out digits. For example, looking something like this: "0002340356705020520304570676574506304502350203503525235230457067805065407405634062052034503250324523..." And go on forever. The question is: If you present an algorithm to an AI and ask: "Does this algorithm have property Z?", then it cannot be trusted to get the answer right for every algorithm it's presented with. It exactly what I asked. Ok. Also, I'd like to correct my previous comment I said incorrectly. I asked whether natural numbers had contemplate set, however I didn't ask about of even or odd, but about the other ones (are there any?). That B can have different properties, right? Is it necessary for A to have B – the subset that contains Z? – What is my interest in here? What about such a group of symbols of a certain language (e.g. some mystery Egyptian messages) that can be perfectly translated without that Z. Is it possible? Do all the groups of objects that are being coded (by a computer) have that set Z? And "Z" I meant here as the set that produces such enumerated combinations of symbols.
I don't think I understand your questions. Are you, for example, asking whether it's possible for any computer language to write an algorithm that has property Z, as in, producing an infinite sequence of digits? No. One could create a very simplistic computer language that doesn't allow for that.
Would it be possible to write an algorithm with property Z in a computer language capable of expressing the Egyptian language? I don't know the Egyptian language, but I do know the Sumerian language, and I would state that a computer language not having that ability, would have limitations a Sumerian wouldn't have. Nothing prevents a Sumerian from starting to recite an infinite number sequences based on some mathematical formula.
And are you asking if there are more examples of complementary sets than that of odd and even numbers? Yes, there is an infinite number of such sets. Here's a random example:
Set 1: 3x (=3, 6, 9, 12, 15, 18,...)
Set 2: 3x1 (=2, 5, 8, 11, 14, 17...)
Set 3: 3x2 (=1, 4, 7, 10, 13, 16...)
Set 4: The combined set 2 and set 3. (=1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17...)
Set 4 is the complementary set of set 1.

